These notes are genuine in appearance, so it's likely real. The cut appears on a number of such bills, sometimes twice, if you look for currency examples on eBay and similar sites. That's called a cross-cut cancel, and at the time the device used to do this was a bank hammer like this one, if not this exact pattern--you can see the four L cuts in the X pattern. Cancellation of currency was, as that indicates, performed to invalidate the bill after its exchange for other fiscal instruments ('funding') so it would not remain in circulation and, presumably, expand the money supply. The reason they did not destroy it was because such bills were to be returned to Treasury for verification and then destruction; in cases like yours, the bill was not destroyed, but its exact path into private hands is harder to trace. Certainly the course of the war would dictate complications in the chain of transfer, but without more information on provenance, it's hard to say where and when that might have happened.

Cross-cutting for cancellation of currency was also common at other times; during the Revolutionary War, the ad hoc money in the colonies was canceled with single cuts in an X pattern, though I'm not sure whether a hammer was used. I've seen a few CSA bills with round cutouts as well, in roughly the same places, but they seem less precise than this X tool.